Question 1

Solve.
-$$\sqrt { 3 x } = 2$$

Accepted Answer

A)  12 
B)  $\frac { 2 } { 3 }$ 
C)  6 
D)  $\frac { 4 } { 3 }$ 
A)  12 
B)  $\frac { 2 } { 3 }$ 
C)  6 
D)  $\frac { 4 } { 3 }$

Question 2

Solve.
-$$\sqrt { 43 - x } = x - 1$$

Accepted Answer

A)  7 
B)  $- 6$ 
C)  $- 6,7$ 
D)  $\varnothing$ 
A)  7 
B)  $- 6$ 
C)  $- 6,7$ 
D)  $\varnothing$

Question 3

Multiply or divide.
-$\frac { \sqrt { - 64 } } { \sqrt { - 8 } }$

Accepted Answer

A)  $2 \sqrt { 2 }$ 
B)  $2 \mathrm { i } \sqrt { 2 }$ 
C)  $- 2 \mathrm { i } \sqrt { 2 }$ 
D)  $- 2 \sqrt { 2 }$ 
A)  $2 \sqrt { 2 }$ 
B)  $2 \mathrm { i } \sqrt { 2 }$ 
C)  $- 2 \mathrm { i } \sqrt { 2 }$ 
D)  $- 2 \sqrt { 2 }$

Question 4

Write with positive exponents. Simplify if possible.
-$27 ^ { - 4 / 3 }$

Accepted Answer

A)  $- \frac { 1 } { 81 }$ 
B)  $\frac { 1 } { 81 }$ 
C)  81 
D)  not a real number 
A)  $- \frac { 1 } { 81 }$ 
B)  $\frac { 1 } { 81 }$ 
C)  81 
D)  not a real number

Question 5

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 2 } { 3 i }$

Accepted Answer

A)  $\frac { 2 } { 3 } \mathrm { i }$ 
B)  $- \frac { 3 } { 2 } \mathrm { i }$ 
C)  $- \frac { 2 } { 3 } \mathrm { i }$ 
D)  $\frac { 3 } { 2 } \mathrm { i }$ 
A)  $\frac { 2 } { 3 } \mathrm { i }$ 
B)  $- \frac { 3 } { 2 } \mathrm { i }$ 
C)  $- \frac { 2 } { 3 } \mathrm { i }$ 
D)  $\frac { 3 } { 2 } \mathrm { i }$

Question 6

Use radical notation to write the expression. Simplify if possible.
-$9 x ^ { 1 / 5 }$

Accepted Answer

A)  $\frac { 9 } { 5 } x$ 
B)  $\sqrt [ 5 ] { 9 \mathrm { x } }$ 
C)  $9 x$ 
D)  $9 \sqrt [ 5 ] { x }$ 
A)  $\frac { 9 } { 5 } x$ 
B)  $\sqrt [ 5 ] { 9 \mathrm { x } }$ 
C)  $9 x$ 
D)  $9 \sqrt [ 5 ] { x }$

Question 7

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 3 ] { \frac { 4 } { 9 x ^ { 2 } } }$

Accepted Answer

A)  $\frac { \sqrt [ 3 ] { 324 x ^ { 2 } } } { 9 x }$ 
B)  $\frac { \sqrt [ 3 ] { 324 x } } { 81 }$ 
C)  $\frac { \sqrt [ 3 ] { 36 x } } { 9 x }$ 
D)  $\frac { \sqrt [ 3 ] { 12 x } } { 3 x }$ 
A)  $\frac { \sqrt [ 3 ] { 324 x ^ { 2 } } } { 9 x }$ 
B)  $\frac { \sqrt [ 3 ] { 324 x } } { 81 }$ 
C)  $\frac { \sqrt [ 3 ] { 36 x } } { 9 x }$ 
D)  $\frac { \sqrt [ 3 ] { 12 x } } { 3 x }$

Question 8

Rationalize the numerator and simplify. Assume all variables represent positive real numbers.
-$$\frac { 5 + \sqrt { 6 } } { 5 - \sqrt { 6 } }$$

Accepted Answer

A)  $\frac { 19 } { 31 + 10 \sqrt { 6 } }$ 
B)  $\frac { 19 } { 31 - 10 \sqrt { 6 } }$ 
C)  $\frac { 31 } { 19 - 10 \sqrt { 6 } }$ 
D)  1 
A)  $\frac { 19 } { 31 + 10 \sqrt { 6 } }$ 
B)  $\frac { 19 } { 31 - 10 \sqrt { 6 } }$ 
C)  $\frac { 31 } { 19 - 10 \sqrt { 6 } }$ 
D)  1

Question 9

Simplify. Assume that all variables represent any real number.
-$$\sqrt { x ^ { 2 } + 18 x + 81 }$$

Accepted Answer

A)  $x + 9$ 
B)  $- x - 9$ 
C)  $| x + 9 |$ 
D)  $- | x + 9 |$ 
A)  $x + 9$ 
B)  $- x - 9$ 
C)  $| x + 9 |$ 
D)  $- | x + 9 |$

Question 10

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$$\sqrt [ 4 ] { 16 } \cdot \sqrt [ 4 ] { 256 n ^ { 4 } }$$

Accepted Answer

A)  $6 n$ 
B)  $- 2 n$ 
C)  $- 8 n$ 
D)  $8 n$ 
A)  $6 n$ 
B)  $- 2 n$ 
C)  $- 8 n$ 
D)  $8 n$

Question 11

Find the root. Assume that all variables represent nonnegative real numbers.
-$\sqrt [ 6 ] { 729 }$

Accepted Answer

A)  3 
B)  27 
C)  9 
D)  not a real number 
A)  3 
B)  27 
C)  9 
D)  not a real number

Question 12

Multiply, and then simplify if possible. Assume all variables represent positive real numbers.
-$( \sqrt { 35 } - \sqrt { 245 } ) ( \sqrt { 7 } + \sqrt { 5 } )$

Accepted Answer

A)  $7 \sqrt { 5 } + 5 \sqrt { 7 } - 7 \sqrt { 35 } - 35$ 
B)  $7 \sqrt { 5 } - 5 \sqrt { 7 } + 7 \sqrt { 35 } + 35$ 
C)  $7 \sqrt { 5 } + 5 \sqrt { 7 } - 7 \sqrt { 35 } + 35$ 
D)  $7 \sqrt { 5 } + 5 \sqrt { 7 } + 7 \sqrt { 35 } + 35$ 
A)  $7 \sqrt { 5 } + 5 \sqrt { 7 } - 7 \sqrt { 35 } - 35$ 
B)  $7 \sqrt { 5 } - 5 \sqrt { 7 } + 7 \sqrt { 35 } + 35$ 
C)  $7 \sqrt { 5 } + 5 \sqrt { 7 } - 7 \sqrt { 35 } + 35$ 
D)  $7 \sqrt { 5 } + 5 \sqrt { 7 } + 7 \sqrt { 35 } + 35$

Question 13

Perform the indicated operation. Write the result in the form a + bi.
-$$2 i ( 8 - 8 i )$$

Accepted Answer

A)  $16 i - 16 i ^ { 2 }$ 
B)  $16 i - 16$ 
C)  $16 + 16 i$ 
D)  $16 i + 16 i ^ { 2 }$ 
A)  $16 i - 16 i ^ { 2 }$ 
B)  $16 i - 16$ 
C)  $16 + 16 i$ 
D)  $16 i + 16 i ^ { 2 }$

Question 14

Perform the indicated operation. Assume that all variables represent positive real numbers.
-$( \sqrt { 6 } - 3 ) ( \sqrt { 5 } - 6 )$

Accepted Answer

A)  $\sqrt { 30 } + 18$ 
B)  $\sqrt { 30 } - 6 \sqrt { 6 } - 3 \sqrt { 5 } + 18$ 
C)  $- 8 \sqrt { 30 } + 18$ 
D)  $\sqrt { 30 } - 9 \sqrt { 5 } + 18$ 
A)  $\sqrt { 30 } + 18$ 
B)  $\sqrt { 30 } - 6 \sqrt { 6 } - 3 \sqrt { 5 } + 18$ 
C)  $- 8 \sqrt { 30 } + 18$ 
D)  $\sqrt { 30 } - 9 \sqrt { 5 } + 18$

Question 15

Solve.
-$$\sqrt [ 3 ] { 3 x + 5 } + 5 = 0$$

Accepted Answer

A)  $- \frac { 130 } { 3 }$ 
B)  $- \frac { 125 } { 3 }$ 
C)  $\frac { 20 } { 3 }$ 
D)  $- \frac { 128 } { 5 }$ 
A)  $- \frac { 130 } { 3 }$ 
B)  $- \frac { 125 } { 3 }$ 
C)  $\frac { 20 } { 3 }$ 
D)  $- \frac { 128 } { 5 }$

Question 16

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 4 } { 4 + i }$

Accepted Answer

A)  $\frac { 16 } { 15 } - \frac { 4 } { 15 } \mathrm { i }$ 
B)  $\frac { 16 } { 15 } + \frac { 4 } { 15 } \mathrm { i }$ 
C)  $\frac { 16 } { 17 } - \frac { 4 } { 17 } \mathrm { i }$ 
D)  $\frac { 16 } { 17 } + \frac { 4 } { 17 } \mathrm { i }$ 
A)  $\frac { 16 } { 15 } - \frac { 4 } { 15 } \mathrm { i }$ 
B)  $\frac { 16 } { 15 } + \frac { 4 } { 15 } \mathrm { i }$ 
C)  $\frac { 16 } { 17 } - \frac { 4 } { 17 } \mathrm { i }$ 
D)  $\frac { 16 } { 17 } + \frac { 4 } { 17 } \mathrm { i }$

Question 17

Find the midpoint of the line segment whose endpoints are given.
-$$( 7 , - 5 ) , ( - 4,5 )$$

Accepted Answer

A)  $( 11 , - 10 )$ 
B)  $\left( \frac { 11 } { 2 } , - 5 \right)$ 
C)  $\left( \frac { 3 } { 2 } , 0 \right)$ 
D)  $( 3,0 )$ 
A)  $( 11 , - 10 )$ 
B)  $\left( \frac { 11 } { 2 } , - 5 \right)$ 
C)  $\left( \frac { 3 } { 2 } , 0 \right)$ 
D)  $( 3,0 )$

Question 18

Multiply, and then simplify if possible. Assume all variables represent positive real numbers.
-$4 \sqrt { 7 } ( \sqrt { 11 } + \sqrt { 7 } )$

Accepted Answer

A)  $4 \sqrt { 11 } + 7$ 
B)  $4 \sqrt { 77 } + 7$ 
C)  $28 \sqrt { 11 } + 28$ 
D)  $4 \sqrt { 77 } + 28$ 
A)  $4 \sqrt { 11 } + 7$ 
B)  $4 \sqrt { 77 } + 7$ 
C)  $28 \sqrt { 11 } + 28$ 
D)  $4 \sqrt { 77 } + 28$

Question 19

Find the power of i.
-$\mathrm { i } ^ { 20 }$

Accepted Answer

A)  $i$ 
B)  $- 1$ 
C)  1 
D)  $- \mathrm { i }$ 
A)  $i$ 
B)  $- 1$ 
C)  1 
D)  $- \mathrm { i }$

Question 20

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 3 } { \sqrt { 125 x } }$

Accepted Answer

A)  $\frac { \sqrt { 15 x } } { 25 x }$ 
B)  $\frac { 3 \sqrt { x } } { 5 x }$ 
C)  $\frac { 3 \sqrt { 5 x } } { 25 x }$ 
D)  $\frac { 3 \sqrt { 125 x } } { 125 x }$ 
A)  $\frac { \sqrt { 15 x } } { 25 x }$ 
B)  $\frac { 3 \sqrt { x } } { 5 x }$ 
C)  $\frac { 3 \sqrt { 5 x } } { 25 x }$ 
D)  $\frac { 3 \sqrt { 125 x } } { 125 x }$

Solve. - $\sqrt { 3 x } = 2$

Solve. - $\sqrt { 43 - x } = x - 1$

Multiply or divide. - $\frac { \sqrt { - 64 } } { \sqrt { - 8 } }$

Write with positive exponents. Simplify if possible. - $27 ^ { - 4 / 3 }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 2 } { 3 i }$

Use radical notation to write the expression. Simplify if possible. - $9 x ^ { 1 / 5 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { \frac { 4 } { 9 x ^ { 2 } } }$

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - $\frac { 5 + \sqrt { 6 } } { 5 - \sqrt { 6 } }$

Simplify. Assume that all variables represent any real number. - $\sqrt { x ^ { 2 } + 18 x + 81 }$

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt [ 4 ] { 16 } \cdot \sqrt [ 4 ] { 256 n ^ { 4 } }$

Find the root. Assume that all variables represent nonnegative real numbers. - $\sqrt [ 6 ] { 729 }$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( \sqrt { 35 } - \sqrt { 245 } ) ( \sqrt { 7 } + \sqrt { 5 } )$

Perform the indicated operation. Write the result in the form a + bi. - $2 i ( 8 - 8 i )$

Perform the indicated operation. Assume that all variables represent positive real numbers. - $( \sqrt { 6 } - 3 ) ( \sqrt { 5 } - 6 )$

Solve. - $\sqrt [ 3 ] { 3 x + 5 } + 5 = 0$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 4 } { 4 + i }$

Find the midpoint of the line segment whose endpoints are given. - $( 7 , - 5 ) , ( - 4,5 )$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $4 \sqrt { 7 } ( \sqrt { 11 } + \sqrt { 7 } )$

Find the power of i. - $\mathrm { i } ^ { 20 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 3 } { \sqrt { 125 x } }$

Real Numbers and Algebraic Expressions

Equations, Inequalities, and Problem Solving

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Exam 7: Rational Exponents, Radicals, and Complex Numbers

Solve. - 3x=2\sqrt { 3 x } = 23x​=2

Solve. - 43−x=x−1\sqrt { 43 - x } = x - 143−x​=x−1

Multiply or divide. - −64−8\frac { \sqrt { - 64 } } { \sqrt { - 8 } }−8​−64​​

Write with positive exponents. Simplify if possible. - 27−4/327 ^ { - 4 / 3 }27−4/3

Perform the indicated operation. Write the result in the form a + bi. - 23i\frac { 2 } { 3 i }3i2​

Use radical notation to write the expression. Simplify if possible. - 9x1/59 x ^ { 1 / 5 }9x1/5

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 49x23\sqrt [ 3 ] { \frac { 4 } { 9 x ^ { 2 } } }39x24​​

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - 5+65−6\frac { 5 + \sqrt { 6 } } { 5 - \sqrt { 6 } }5−6​5+6​​

Simplify. Assume that all variables represent any real number. - x2+18x+81\sqrt { x ^ { 2 } + 18 x + 81 }x2+18x+81​

Use the product rule to multiply. Assume all variables represent positive real numbers. - 164⋅256n44\sqrt [ 4 ] { 16 } \cdot \sqrt [ 4 ] { 256 n ^ { 4 } }416​⋅4256n4​

Find the root. Assume that all variables represent nonnegative real numbers. - 7296\sqrt [ 6 ] { 729 }6729​

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - (35−245)(7+5)( \sqrt { 35 } - \sqrt { 245 } ) ( \sqrt { 7 } + \sqrt { 5 } )(35​−245​)(7​+5​)

Perform the indicated operation. Write the result in the form a + bi. - 2i(8−8i)2 i ( 8 - 8 i )2i(8−8i)

Perform the indicated operation. Assume that all variables represent positive real numbers. - (6−3)(5−6)( \sqrt { 6 } - 3 ) ( \sqrt { 5 } - 6 )(6​−3)(5​−6)

Solve. - 3x+53+5=0\sqrt [ 3 ] { 3 x + 5 } + 5 = 033x+5​+5=0

Perform the indicated operation. Write the result in the form a + bi. - 44+i\frac { 4 } { 4 + i }4+i4​

Find the midpoint of the line segment whose endpoints are given. - (7,−5),(−4,5)( 7 , - 5 ) , ( - 4,5 )(7,−5),(−4,5)

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - 47(11+7)4 \sqrt { 7 } ( \sqrt { 11 } + \sqrt { 7 } )47​(11​+7​)

Find the power of i. - i20\mathrm { i } ^ { 20 }i20

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 3125x\frac { 3 } { \sqrt { 125 x } }125x​3​